

A326969


Number of sets of subsets of {1..n} whose dual is a weak antichain.


8




OFFSET

0,1


COMMENTS

The dual of a set of subsets has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.


LINKS

Table of n, a(n) for n=0..4.


FORMULA

a(n) = 2 * A326968(n).
a(n) = 2 * Sum_{k = 0..n} binomial(n, k) * A326970(k).


EXAMPLE

The a(0) = 2 through a(2) = 12 sets of subsets:
{} {} {}
{{}} {{}} {{}}
{{1}} {{1}}
{{},{1}} {{2}}
{{1,2}}
{{},{1}}
{{},{2}}
{{1},{2}}
{{},{1,2}}
{{},{1},{2}}
{{1},{2},{1,2}}
{{},{1},{2},{1,2}}


MATHEMATICA

dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Table[Length[Select[Subsets[Subsets[Range[n]]], stableQ[dual[#], SubsetQ]&]], {n, 0, 3}]


CROSSREFS

Sets of subsets whose dual is strict are A326941.
The BIInumbers of setsystems whose dual is a weak antichain are A326966.
Sets of subsets whose dual is a (strict) antichain are A326967.
The case without empty edges is A326968.
Cf. A001146, A059052, A326951, A326970, A326971, A326975, A326978.
Sequence in context: A326950 A001696 A276534 * A304986 A013333 A154882
Adjacent sequences: A326966 A326967 A326968 * A326970 A326971 A326972


KEYWORD

nonn,more


AUTHOR

Gus Wiseman, Aug 10 2019


STATUS

approved



